Second Chern class $c_2$ of $SU(n)$ bundle over $S^4$ v.s. $\pi_3(SU(n))=\mathbb{Z}$

Question

Let us compare the four statements:

1. Consider the $SU(2)$ fiber over the $S^4$. Such that this $SU(2) = S^3$ fiber over $S^4$ gives a second Chern class $c_2$ on the $S^4$ with $c_2=1$.

2. Consider the $S^3$ fiber over $S^4$ as a Hopf fibration, so that we have a $S^7$ constructed out of $S^3 \hookrightarrow S^7 \to S^4$.

3. Consider the classification of the homotopy map between the $SU(2)=S^3$ and the target space as the $S^3$ boundary of the flat $\mathbb{R}^4$, as $\pi_3(S^3)=\mathbb{Z}$. To compare with the first two statements, we take the unit integer class 1 in $\pi_3(S^3)=\mathbb{Z}$. I think there is a correspondence of this map to the above two constructions.

4. Consider the homotopy class map between $M=S^4$ and $BSU(2)=\mathbf{HP}^\infty$, so that $[M, BSU(2)]=[S^4,\mathbf{HP}^\infty] = [S^4, K(\mathbb{Q},4)]= [S^4, K(\mathbb{Z},4)_{\mathbb{Q}}]$ (see https://en.wikipedia.org/wiki/Quaternionic_projective_space) which is $H^4(S^4,\mathbb{Z})=\mathbb{Z}$ (?). To compare with the first three statements, we take the unit integer class 1 in $H^4(S^4,\mathbb{Z})=\mathbb{Z}$. I think there is a correspondence of this map to the above two constructions.

Question 1 --- How to show or prove that the above four statements are the same or related constructions? Especially the first two statements?

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Question 2 --- I think this statement can be generalized to the second Chern class $c_2$ of $SU(n)$ bundle over $S^4$ v.s. $\pi_3(SU(n))=\mathbb{Z}$. How does it go? Here is an attempt.

1. Consider the $SU(n)$ fiber over the $S^4$. Such that this $SU(n)$ fiber over $S^4$ gives a second Chern class $c_2$ on the $S^4$ with $c_2=1$.

2. Consider the $SU(n)$ fiber over $S^4$ as a Hopf fibration, so that we have a construction out of $SU(n) \hookrightarrow ??? \to S^4$.

3. Consider the classification of the homotopy map between the $SU(n)$ and the target space as the $S^3$ boundary of the flat $\mathbb{R}^4$, as $\pi_3(SU(n))=\mathbb{Z}$ for $n \geq 2$. To compare with the first two statements, we take the unit integer class 1 in $\pi_3(SU(n))=\mathbb{Z}$. I think there is a correspondence of this map to the above two constructions.

4. Consider the homotopy class map between $M=S^4$ and $BSU(n)$, so that $[M, BSU(n)]=[S^4, BSU(n)]=?$ which may be related to $H^4(S^4,\mathbb{Z})=\mathbb{Z}$ (?). To compare with the first three statements, we take the unit integer class 1 in $H^4(S^4,\mathbb{Z})=\mathbb{Z}$. I think there is a correspondence of this map to the above two constructions.